Complex Numbers Chapter 2 66 repeatedly, and find exactly 4 fourth roots. We can read the values of 64 in rectangular form from Figure 10.3: V-64=+2 ± 2i (all four combinations of ± signs) or we can calculate them as in Example 2, or we can solve the equation 24 = -64 by computer Example 4. Find and plot all values of -81. The polar coordinates of -8i are r = 8, 0 270°360°k = 3T/2 2Tk. Then the polar coordinates of -8i are 270+360°k = 45°+60°k or (10.5) In Figure 10.4, we sketch a circle of radius V2. On it we plot the point at 45° and then plot the rest of the 6 equally spaced points 60° apart. To find the roots in rectangular coordinates, we need to find all the values of r(cos 0 i sin 0) with r and 0 given by (10.5). We can do this one root at a time or more simply by using a computer to solve the equation 26 = -8i. We find (see Problem 33) Figure 10.4 1 1.366 -0.366i, 0.366-1.366i}. Summary In each of the preceding examples, our steps in finding Vrei were (a) Find the polar coordinates of the roots: Take the nth root of r and divide e+2kT by n. (b) Make a sketch: Draw a circle of radius, plot the root with angle 0/n, and then plot the rest of the n roots around the circle equally spaced 2T/n apart. Note that we have now essentially solved the problem. From the sketch you can see the approximate rectangular coordinates of the roots and check your answers in (c). Since this sketch is quick and easy to do, it is worthwhile even if you use a computer to do part (c) (c) Find the r+iy coordinates of the roots by one of the methods in the examples. If you are using a computer, you may want to make a computer plot of the roots which should be a perfected copy of your sketch in (b). PROBLEMS, SECTION 10 Follow steps (a), (b), (c) above to find all the values of the indicated roots VT 2. 27 VT 1. 3. VI6 5. 4. 6. 7. 32 11. S 10. 12. Section 11 The Exponential and Trigonometric Functions 67 64 14 13. 15. 16 17 19 2+2iv3 20. 21
Complex Numbers Chapter 2 66 repeatedly, and find exactly 4 fourth roots. We can read the values of 64 in rectangular form from Figure 10.3: V-64=+2 ± 2i (all four combinations of ± signs) or we can calculate them as in Example 2, or we can solve the equation 24 = -64 by computer Example 4. Find and plot all values of -81. The polar coordinates of -8i are r = 8, 0 270°360°k = 3T/2 2Tk. Then the polar coordinates of -8i are 270+360°k = 45°+60°k or (10.5) In Figure 10.4, we sketch a circle of radius V2. On it we plot the point at 45° and then plot the rest of the 6 equally spaced points 60° apart. To find the roots in rectangular coordinates, we need to find all the values of r(cos 0 i sin 0) with r and 0 given by (10.5). We can do this one root at a time or more simply by using a computer to solve the equation 26 = -8i. We find (see Problem 33) Figure 10.4 1 1.366 -0.366i, 0.366-1.366i}. Summary In each of the preceding examples, our steps in finding Vrei were (a) Find the polar coordinates of the roots: Take the nth root of r and divide e+2kT by n. (b) Make a sketch: Draw a circle of radius, plot the root with angle 0/n, and then plot the rest of the n roots around the circle equally spaced 2T/n apart. Note that we have now essentially solved the problem. From the sketch you can see the approximate rectangular coordinates of the roots and check your answers in (c). Since this sketch is quick and easy to do, it is worthwhile even if you use a computer to do part (c) (c) Find the r+iy coordinates of the roots by one of the methods in the examples. If you are using a computer, you may want to make a computer plot of the roots which should be a perfected copy of your sketch in (b). PROBLEMS, SECTION 10 Follow steps (a), (b), (c) above to find all the values of the indicated roots VT 2. 27 VT 1. 3. VI6 5. 4. 6. 7. 32 11. S 10. 12. Section 11 The Exponential and Trigonometric Functions 67 64 14 13. 15. 16 17 19 2+2iv3 20. 21
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Topic Video
Question
Problems, Section 10 number 3,4,7,9. Please answer the questions
Expert Solution
Trending now
This is a popular solution!
Step by step
Solved in 9 steps with 9 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,